Advertisement

Basic Graph Theory

  • Dieter Jungnickel
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 5)

Abstract

Graph theory began in 1736 when Leonhard Euler solved the well-known Königsberg bridge problem. This problem asked for a circular walk through the town of Königsberg (now Kaliningrad) in such a way as to cross over each of the seven bridges spanning the river Pregel once, and only once. Euler realized that the precise shapes of the island and the other three territories involved are not important; the solvability depends only on their connection properties. This led to the abstract notion of a graph. Actually, Euler proved much more: he gave a necessary and sufficient condition for an arbitrary graph to admit such a circular walk. His theorem is one of the highlights in the introductory Chap. 1, where we deal with some of the most fundamental notions in graph theory: paths, cycles, connectedness, 1-factors, trees, Euler tours and Hamiltonian cycles, the travelling salesman problem, drawing graphs in the plane, and directed graphs. We will also see a first application, namely setting up a schedule for a tournament, say in soccer or basketball, where each of the 2n participating teams should play against each of the other teams exactly twice, once at home and once away.

Keywords

Planar Graph Hamiltonian Cycle Euler Tour Hamiltonian Graph Start Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [Aig84]
    Aigner, M.: Graphentheorie. Eine Entwicklung aus dem 4-Farben-Problem. Teubner, Stuttgart (1984) zbMATHGoogle Scholar
  2. [And90]
    Anderson, I.: Combinatorial Designs: Construction Methods. Ellis Horwood, Chichester (1990) zbMATHGoogle Scholar
  3. [And97]
    Anderson, I.: Combinatorial Designs and Tournaments. Oxford University Press, Oxford (1997) zbMATHGoogle Scholar
  4. [BalCo87]
    Ball, W.W.R., Coxeter, H.S.M.: Mathematical Recreations and Essays, 13th edn. Dover, New York (1987) Google Scholar
  5. [BanGu09]
    Bang-Jensen, J., Gutin, G.Z.: Digraphs, 2nd edn. Springer, London (2009) zbMATHCrossRefGoogle Scholar
  6. [BarSa95]
    Barnes, T.M., Savage, C.D.: A recurrence for counting graphical partitions. Electron. J. Comb. 2, # R 11 (1995) Google Scholar
  7. [Ber73]
    Berge, C.: Graphs and Hypergraphs. North Holland, Amsterdam (1973) zbMATHGoogle Scholar
  8. [Ber78]
    Bermond, J.C.: Hamiltonian graphs. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory, pp. 127–167. Academic Press, New York (1978) Google Scholar
  9. [BetJL99]
    Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn. vols. 1 and 2. Cambridge University Press, Cambridge (1999) CrossRefGoogle Scholar
  10. [BigLW76]
    Biggs, N.L., Lloyd, E.K., Wilson, R.J.: Graph Theory 1736–1936. Oxford University Press, Oxford (1976) zbMATHGoogle Scholar
  11. [BoeTi80]
    Boesch, F., Tindell, R.: Robbins theorem for mixed multigraphs. Am. Math. Mon. 87, 716–719 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  12. [BonCh76]
    Bondy, J.A., Chvátal, V.: A method in graph theory. Discrete Math. 15, 111–135 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Bor60]
    Borchardt, C.W.: Über eine der Interpolation entsprechende Darstellung der Eliminationsresultante. J. Reine Angew. Math. 57, 111–121 (1860) zbMATHCrossRefGoogle Scholar
  14. [CamLi91]
    Cameron, P.J., van Lint, J.H.: Designs, Graphs, Codes and Their Links. Cambridge University Press, Cambridge (1991) zbMATHCrossRefGoogle Scholar
  15. [Cay89]
    Cayley, A.: A theorem on trees. Q. J. Math. 23, 376–378 (1889) Google Scholar
  16. [ChuGT85]
    Chung, F.R.K., Garey, M.R., Tarjan, R.E.: Strongly connected orientations of mixed multigraphs. Networks 15, 477–484 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  17. [Chv85]
    Chvátal, V.: Hamiltonian cycles. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) The Travelling Salesman Problem, pp. 403–429. Wiley, New York (1985) Google Scholar
  18. [ChvTh78]
    Chvátal, V., Thomasson, C.: Distances in orientations of graphs. J. Comb. Theory, Ser. B 24, 61–75 (1978) CrossRefGoogle Scholar
  19. [ConHMW92]
    Conrad, A., Hindrichs, T., Morsy, H., Wegener, I.: Wie es einem Springer gelingt, Schachbretter von beliebiger Größe zwischen beliebig vorgegebenen Anfangs- und Endfeldern vollständig abzureiten. Spektrum der Wiss. 10–14 (1992) Google Scholar
  20. [ConHMW94]
    Conrad, A., Hindrichs, T., Morsy, H., Wegener, I.: Solution of the knight’s Hamiltonian path problem on chessboards. Discrete Appl. Math. 50, 125–134 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  21. [Cox73]
    Coxeter, H.M.S.: Regular Polytopes, 3rd edn. Dover, New York (1973) Google Scholar
  22. [deW80]
    de Werra, D.: Geography, games and graphs. Discrete Appl. Math. 2, 327–337 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  23. [deW81]
    de Werra, D.: Scheduling in sports. Ann. Discrete Math. 11, 381–395 (1981) zbMATHGoogle Scholar
  24. [deW82]
    de Werra, D.: Minimizing irregularities in sports schedules using graph theory. Discrete Appl. Math. 4, 217–226 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  25. [deW88]
    de Werra, D.: Some models of graphs for scheduling sports competitions. Discrete Appl. Math. 21, 47–65 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  26. [deWJM90]
    de Werra, D., Jacot-Descombes, L., Masson, P.: A constrained sports scheduling problem. Discrete Appl. Math. 26, 41–49 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Die10]
    Diestel, R.: Graph Theory, 4th edn. Springer, Berlin (2010) CrossRefGoogle Scholar
  28. [Dir52]
    Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952) MathSciNetzbMATHCrossRefGoogle Scholar
  29. [Eul36]
    Euler, L.: Solutio problematis ad geometriam situs pertinentis. Comment. Acad. Sci. Imp. Petropol. 8, 128–140 (1736) Google Scholar
  30. [Eul52/53]
    Euler, L.: Demonstratio nonnullorum insignium proprietatum quibus solida hadris planis inclusa sunt praedita. Novi Comment. Acad. Sci. Petropol. 4, 140–160 (1752/1753) Google Scholar
  31. [Eul66]
    Euler, L.: Solution d’une question curieuse qui ne paroit soumise à aucune analyse. Mém. Acad. R. Sci. Belles Lettres, Année 1759 15, 310–337 (1766) Google Scholar
  32. [Fle83]
    Fleischner, H.: Eulerian graphs. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory 2, pp. 17–53. Academic Press, New York (1983) Google Scholar
  33. [Fle90]
    Fleischner, H.: Eulerian Graphs and Related Topics, Part 1, vol. 1. North Holland, Amsterdam (1990) zbMATHGoogle Scholar
  34. [Fle91]
    Fleischner, H.: Eulerian Graphs and Related Topics, Part 1, vol. 2. North Holland, Amsterdam (1991) zbMATHGoogle Scholar
  35. [GonMi84]
    Gondran, M., Minoux, N.: Graphs and Algorithms. Wiley, New York (1984) zbMATHGoogle Scholar
  36. [GouJa83]
    Goulden, I.P., Jackson, D.M.: Combinatorial Enumeration. Wiley, New York (1983) zbMATHGoogle Scholar
  37. [GriRo96]
    Griggs, T., Rosa, A.: A tour of European soccer schedules, or testing the popularity of GK 2n. Bull. Inst. Comb. Appl. 18, 65–68 (1996) MathSciNetzbMATHGoogle Scholar
  38. [Har69]
    Harary, F.: Graph Theory. Addison Wesley, Reading (1969) Google Scholar
  39. [HarTu65]
    Harary, F., Tutte, W.T.: A dual form of Kuratowski’s theorem. Can. Math. Bull. 8, 17–20 and 173 (1965) Google Scholar
  40. [Jel03]
    Jelliss, G.: Knight’s Tour Notes (2003). http://www.ktn.freeuk.com/index.htm
  41. [Kir47]
    Kirkman, T.P.: On a problem in combinatorics. Camb. Dublin Math. J. 2, 191–204 (1847) Google Scholar
  42. [Kur30]
    Kuratowski, K.: Sur le problème des courbes gauches en topologie. Fundam. Math. 15, 271–283 (1930) zbMATHGoogle Scholar
  43. [Lam87]
    Lamken, E.: A note on partitioned balanced tournament designs. Ars Comb. 24, 5–16 (1987) MathSciNetzbMATHGoogle Scholar
  44. [LamVa87]
    Lamken, E., Vanstone, S.A.: The existence of partitioned balanced tournament designs. Ann. Discrete Math. 34, 339–352 (1987) MathSciNetGoogle Scholar
  45. [LamVa89]
    Lamken, E., Vanstone, S.A.: Balanced tournament designs and related topics. Discrete Math. 77, 159–176 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  46. [Las72]
    Las Vergnas, M.: Problèmes de couplages et problèmes hamiltoniens en théorie des graphes. Dissertation, Universitè de Paris VI (1972) Google Scholar
  47. [LesOe86]
    Lesniak, L., Oellermann, O.R.: An Eulerian exposition. J. Graph Theory 10, 277–297 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  48. [Lue89]
    Lüneburg, H.: Tools and Fundamental Constructions of Combinatorial Mathematics. Bibliographisches Institut, Mannheim (1989) zbMATHGoogle Scholar
  49. [MenRo85]
    Mendelsohn, E., Rosa, A.: One-factorizations of the complete graph—a survey. J. Graph Theory 9, 43–65 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  50. [NisCh88]
    Nishizeki, T., Chiba, N.: Planar Graphs: Theory and Algorithms. North Holland, Amsterdam (1988) zbMATHGoogle Scholar
  51. [Ore60]
    Ore, O.: Note on Hamiltonian circuits. Am. Math. Mon. 67, 55 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  52. [Oza25]
    Ozanam, J.: Récréations Mathématiques et Physiques, vol. 1. Claude Jombert, Paris (1725) Google Scholar
  53. [Pet98]
    Petersen, J.: Sur le théorème de Tait. L’Intermédiaire Math. 5, 225–227 (1898) Google Scholar
  54. [Pri96]
    Prisner, E.: Line graphs and generalizations—a survey. Congr. Numer. 116, 193–229 (1996) MathSciNetzbMATHGoogle Scholar
  55. [Pru18]
    Prüfer, H.: Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. (3) 27, 142–144 (1918) zbMATHGoogle Scholar
  56. [Ren59]
    Rényi, A.: Some remarks on the theory of trees. Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 4, 73–85 (1959) zbMATHGoogle Scholar
  57. [Rob39]
    Robbins, H.: A theorem on graphs with an application to a problem of traffic control. Am. Math. Mon. 46, 281–283 (1939) MathSciNetCrossRefGoogle Scholar
  58. [RobXu88]
    Roberts, F.S., Xu, Y.: On the optimal strongly connected orientations of city street graphs I: Large grids. SIAM J. Discrete Math. 1, 199–222 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  59. [Schr80]
    Schreuder, J.A.M.: Constructing timetables for sport competitions. Math. Program. Stud. 13, 58–67 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  60. [Schr92]
    Schreuder, J.A.M.: Combinatorial aspects of construction of competition Dutch professional football leagues. Discrete Appl. Math. 35, 301–312 (1992) zbMATHCrossRefGoogle Scholar
  61. [Schw91]
    Schwenk, A.J.: Which rectangular chessboards have a knight’s tour? Math. Mag. 64, 325–332 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  62. [SieHo91]
    Sierksma, G., Hoogeveen, H.: Seven criteria for integer sequences being graphic. J. Graph Theory 15, 223–231 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  63. [Sta86]
    Stanley, R.P.: Enumerative Combinatorics, vol. 1. Wadsworth & Brooks/Cole, Monterey (1986) zbMATHGoogle Scholar
  64. [Sta99]
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999) CrossRefGoogle Scholar
  65. [Tak90a]
    Takács, L.: On Cayley’s formula for counting forests. J. Comb. Theory, Ser. A 53, 321–323 (1990) zbMATHCrossRefGoogle Scholar
  66. [Tak90b]
    Takács, L.: On the number of distinct forests. SIAM J. Discrete Math. 3, 574–581 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  67. [Tho81]
    Thomassen, C.: Kuratowski’s theorem. J. Graph Theory 5, 225–241 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  68. [Wag36]
    Wagner, K.: Bemerkungen zum Vierfarbenproblem. Jahresber. DMV 46, 26–32 (1936) Google Scholar
  69. [Wag37]
    Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114, 170–190 (1937) CrossRefGoogle Scholar
  70. [Wal92]
    Wallis, W.D.: One-factorizations of the complete graph. In: Dinitz, J.H., Stinson, D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 593–639. Wiley, New York (1992) Google Scholar
  71. [Wil86]
    Wilson, R.J.: An Eulerian trail through Königsberg. J. Graph Theory 10, 265–275 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  72. [Wil89]
    Wilson, R.J.: A brief history of Hamiltonian graphs. Ann. Discrete Math. 41, 487–496 (1989) CrossRefGoogle Scholar
  73. [Yap86]
    Yap, H.P.: Some Topics in Graph Theory. Cambridge University Press, Cambridge (1986) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Dieter Jungnickel
    • 1
  1. 1.Institut für MathematikUniversität AugsburgAugsburgGermany

Personalised recommendations